30 Dec A wonderfully confusing little problem (LEONARD MLODINOW) | Part A’
As newspaper columns go, Parade magazine’s “Ask Marilyn” has to be considered a smashing success. Distributed in 350 newspapers and boasting a combined circulation of nearly 36 million, the
question-and-answer column originated in 1986 and is still going strong.
One might ask, especially if one knows a little something about mathematics and science, “Who is this guru Marilyn?”
Well, Marilyn is Marilyn vos Savant, famous for being listed for years in the Guinness World Records Hall of Fame as the person with the world’s highest recorded IQ (228). She is also famous for being married to Robert Jarvik, inventor of the Jarvik artificial heart.
Marilyn, is most famous for her response to the following question, which appeared in her column one Sunday in September 1990 (I have altered the wording slightly):
Suppose the contestants on a game show are given the choice of three doors: Behind one door is a car; behind the others, goats. After a contestant picks a door, the host, who knows what’s behind all the doors, opens one of the unchosen doors, which reveals a goat. He then says to the contestant, “Do you want to switch to the other unopened door?”
Is it to the contestant’s advantage to make the switch?’
This issue has immortalized both Marilyn and Let’s Make a Deal because of the vehemence with which Marilyn vos Savant’s readers responded to the column.
After all, it appears to be a pretty silly question. Two doors are available—open one and you win; open the other and you lose—so it seems self-evident that whether you change your choice
or not, your chances of winning are 50/50. What could be simpler?
The thing is, Marilyn said in her column that it is better to switch.
Despite the public’s much-heralded lethargy when it comes to mathematical issues, Marilyn’s readers reacted as if she’d advocated ceding California back to Mexico. Her denial of the obvious brought her an avalanche of mail, 10,000 letters by her estimate. If you ask the American people whether they agree that plants create the oxygen in the air, light travels faster than sound, or you cannot make radioactive milk safe by boiling it, you will get double-digit disagreement
in each case (13 percent, 24 percent, and 35 percent, respectively).
But on this issue, Americans were united: 92 percent agreed Marilyn was wrong.
Many readers seemed to feel let down. How could a person they trusted on such a broad range of issues be confused by such a simple question?
Was her mistake a symbol of the woeful ignorance of the American people?
Almost 1,000 PhDs wrote in, many of them math professors, who seemed to be especially irate.
“You blew it,” wrote a mathematician from George Mason University:
Let me explain: If one door is shown to be a loser, that information changes the probability of either remaining choice— neither of which has any reason to be more likely—to 1⁄2. As a professional mathematician, I’m very concerned with the general public’s lack of mathematical skills. Please help by confessing your error and, in the future, being more careful.
From Dickinson State University came this:
“I am in shock that after being corrected by at least three mathematicians, you still do not see your mistake.”
From Georgetown: “How many irate mathematicians are needed to change your mind?”
And someone from the U.S. Army Research Institute remarked, “If all those PhDs are wrong the country would be in serious trouble.”
Responses continued in such great numbers and for such a long time that after devoting quite a bit of column space to the issue, Marilyn decided she would no longer address it.
The army PhD who wrote in may have been correct that if all those PhDs were wrong, it would be a sign of trouble. But Marilyn was correct.
When told of this, Paul Erdös, one of the leading mathematicians of the twentieth century, said, “That’s impossible.” Then, when presented with a formal mathematical proof of the correct answer, he still didn’t believe it and grew angry.
Only after a colleague arranged for a computer simulation in which Erdös watched hundreds of trials that came out 2 to 1 in favor of switching did Erdös concede he was wrong.
How can something that seems so obvious be wrong?
In the words of a Harvard professor who specializes in probability and statistics, “Our brains are just not wired to do probability problems very well.”
But the Monty Hall problem is one of those that can be solved without any specialized mathematical knowledge. You don’t need calculus, geometry, algebra, or even amphetamines, which Erdös was reportedly fond of taking.
All you need is a basic understanding of how probability works and the law of the sample space, that framework for analyzing chance situations that was first put on paper in the sixteenth century by Gerolamo Cardano.
The Drunkard’s Walk